Path Puzzles: Discrete Tomography with a Path Constraint is Hard

We prove that path puzzles with complete row and column information--or equivalently, 2D orthogonal discrete tomography with Hamiltonicity constraint--are strongly NP-complete, ASP-complete, and #P-complete. Along the way, we newly establish ASP-completeness and #P-completeness for 3-Dimensional Matching and Numerical 3-Dimensional Matching.Comment: 16 pages, 8 figures. Revised proof of Theorem 2.4. 2-page abstract appeared in Abstracts from the 20th Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCGGG 2017

Gas Sorption and Luminescence Properties of a Terbium(III)-Phosphine Oxide Coordination Material with Two-Dimensional Pore Topology

The structure, stability, gas sorption properties and luminescence behaviour of a new lanthanide-phosphine oxide coordination material are reported. The polymer PCM-15 is based on Tb(III) and tris(p-carboxylated) triphenylphosphine oxide and has a 5,5-connected net topology. It exhibits an infinite three-dimensional structure that incorporates an open, two-dimensional pore structure. The material is thermally robust and remains crystalline under high vacuum at 150 degrees C. When desolvated, the solid has a CO2 BET surface area of 1187 m(2) g(-1) and shows the highest reported uptake of both O-2 and H-2 at 77 K and 1 bar for a lanthanide-based coordination polymer. Isolated Tb(III) centres in the as-synthesized polymer exhibit moderate photoluminescence. However, upon removal of coordinated OH2 ligands, the luminescence intensity was found to approximately double; this process was reversible. Thus, the Tb(III) centre was used as a probe to detect directly the desolvation and resolvation of the polymer.Welch Foundation F-1738, F-1631National Science Foundation 0741973, CHE-0847763Chemistr

Chemical Time Bombs: Linkages to Scenarios of Socioeconomic Development (CTB Basic Document 2)

The definition of a chemical time bomb (CTB), as provided in the first document of this series is "a concept that refers to a chain of events resulting in the delayed and sudden occurrence of harmful effects due to the mobilization of chemicals stored in soils and sediments in response to slow alterations of the environment." The theme of this second report was conceived at a workshop in the Netherlands in 1990. It was decided that chemical time bombs must be understood not only in terms of how they are triggered in the environment, but also in terms of the anthropogenic activities that are linked to the triggers. For example, a change in redox potential is a CTB trigger, and activities such as draining of wetlands an implementing sewage treatment have a major influence on redox potential. Thus, this report attempts to connect specific human activities to environmental disturbances that can stimulate CTNB phenomena. These connections are made for a range of activities, and matrices linking activities to effects are presented. The analysis is taken a step further by constructing scenarios, of land-use changes for example, and assessing their impacts with respect to CTBs. Thus, scenarios are used here not as a way of predicting the future, but rather for the purpose of presenting possible alternatives against which the risk of CTB events can be assessed. This publication is the second in a series of IIASA publications on Chemical Time Bombs. The first, entitled "Chemical Time Bombs: Definition, Concepts, and Examples," was published in 1991. The next publication in the series will discuss CTBs in landfills and contaminated lands

Conflict-Free Coloring of Planar Graphs

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number chi_CF(G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N[v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N(v), for which vertex v is not a member of its neighborhood. For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. We also give a complete characterization of the computational complexity of conflict-free coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G, but polynomial for outerplanar graphs. Furthermore, deciding whether chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for outerplanar graphs. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs. For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general} planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on Discrete Mathematics) of extended abstract that appears in Proceeedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pp. 1951-196